math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 8.5s

Alternatives: 16

Speedup: 1.0×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp(-im) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp(-im) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

4. Final simplification 100.0%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right) \]

Alternative 2: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 170000000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 8.6 \cdot 10^{+74}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 170000000.0)
   (*
    (* 0.5 (sin re))
    (+ (+ 2.0 (* im im)) (* (pow im 4.0) 0.08333333333333333)))
   (if (<= im 8.6e+74)
     (* (+ (exp (- im)) (exp im)) (* 0.5 re))
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 170000000.0) {
		tmp = (0.5 * sin(re)) * ((2.0 + (im * im)) + (pow(im, 4.0) * 0.08333333333333333));
	} else if (im <= 8.6e+74) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 170000000.0d0) then
        tmp = (0.5d0 * sin(re)) * ((2.0d0 + (im * im)) + ((im ** 4.0d0) * 0.08333333333333333d0))
    else if (im <= 8.6d+74) then
        tmp = (exp(-im) + exp(im)) * (0.5d0 * re)
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 170000000.0) {
		tmp = (0.5 * Math.sin(re)) * ((2.0 + (im * im)) + (Math.pow(im, 4.0) * 0.08333333333333333));
	} else if (im <= 8.6e+74) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 170000000.0:
		tmp = (0.5 * math.sin(re)) * ((2.0 + (im * im)) + (math.pow(im, 4.0) * 0.08333333333333333))
	elif im <= 8.6e+74:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 170000000.0)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(Float64(2.0 + Float64(im * im)) + Float64((im ^ 4.0) * 0.08333333333333333)));
	elseif (im <= 8.6e+74)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 170000000.0)
		tmp = (0.5 * sin(re)) * ((2.0 + (im * im)) + ((im ^ 4.0) * 0.08333333333333333));
	elseif (im <= 8.6e+74)
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 170000000.0], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.6e+74], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.7e8

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 92.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   6. Simplified 92.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   if 1.7e8 < im < 8.60000000000000001e74

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in re around 0 85.7%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   if 8.60000000000000001e74 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 98.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   6. Simplified 98.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   7. Taylor expanded in im around inf 98.2%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 98.2%

         \[\leadsto 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)} \]

   9. Simplified 98.2%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 170000000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)\\ \mathbf{elif}\;im \leq 8.6 \cdot 10^{+74}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 3: 86.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 170000000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 8.6 \cdot 10^{+74}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 170000000.0)
   (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
   (if (<= im 8.6e+74)
     (* (+ (exp (- im)) (exp im)) (* 0.5 re))
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 170000000.0) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else if (im <= 8.6e+74) {
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 170000000.0d0) then
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    else if (im <= 8.6d+74) then
        tmp = (exp(-im) + exp(im)) * (0.5d0 * re)
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 170000000.0) {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	} else if (im <= 8.6e+74) {
		tmp = (Math.exp(-im) + Math.exp(im)) * (0.5 * re);
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 170000000.0:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	elif im <= 8.6e+74:
		tmp = (math.exp(-im) + math.exp(im)) * (0.5 * re)
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 170000000.0)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 8.6e+74)
		tmp = Float64(Float64(exp(Float64(-im)) + exp(im)) * Float64(0.5 * re));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 170000000.0)
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	elseif (im <= 8.6e+74)
		tmp = (exp(-im) + exp(im)) * (0.5 * re);
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 170000000.0], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.6e+74], N[(N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.7e8

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 84.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

   6. Simplified 84.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   if 1.7e8 < im < 8.60000000000000001e74

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in re around 0 85.7%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   if 8.60000000000000001e74 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 98.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   6. Simplified 98.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   7. Taylor expanded in im around inf 98.2%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 98.2%

         \[\leadsto 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)} \]

   9. Simplified 98.2%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 170000000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 8.6 \cdot 10^{+74}:\\ \;\;\;\;\left(e^{-im} + e^{im}\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 4: 83.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 580:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 580.0)
   (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
   (if (<= im 1.16e+77)
     (+
      0.08333333333333333
      (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 580.0) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else if (im <= 1.16e+77) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 580.0d0) then
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    else if (im <= 1.16d+77) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 580.0) {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	} else if (im <= 1.16e+77) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 580.0:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	elif im <= 1.16e+77:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 580.0)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.16e+77)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 580.0)
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	elseif (im <= 1.16e+77)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 580.0], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.16e+77], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 580

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 84.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

   6. Simplified 84.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   if 580 < im < 1.1600000000000001e77

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   5. Applied egg-rr 14.5%

      \[\leadsto \color{blue}{{\left(\sin re \cdot -2\right)}^{-2}} \]

   6. Taylor expanded in re around 0 38.8%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.25 \cdot \frac{1}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 38.8%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      2. metadata-eval 38.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{\color{blue}{0.25}}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      3. unpow2 38.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{\color{blue}{re \cdot re}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      4. *-commutative 38.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{{re}^{2} \cdot 0.016666666666666666}\right) \]

      5. unpow2 38.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666\right) \]

   8. Simplified 38.8%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]

   if 1.1600000000000001e77 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   6. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 580:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 5: 85.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 120:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 120.0)
   (* (* 0.5 (sin re)) (+ 2.0 (* im im)))
   (if (<= im 1.16e+77)
     (* 0.5 (+ (exp (- im)) (exp im)))
     (* 0.041666666666666664 (* (sin re) (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 120.0) {
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	} else if (im <= 1.16e+77) {
		tmp = 0.5 * (exp(-im) + exp(im));
	} else {
		tmp = 0.041666666666666664 * (sin(re) * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 120.0d0) then
        tmp = (0.5d0 * sin(re)) * (2.0d0 + (im * im))
    else if (im <= 1.16d+77) then
        tmp = 0.5d0 * (exp(-im) + exp(im))
    else
        tmp = 0.041666666666666664d0 * (sin(re) * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 120.0) {
		tmp = (0.5 * Math.sin(re)) * (2.0 + (im * im));
	} else if (im <= 1.16e+77) {
		tmp = 0.5 * (Math.exp(-im) + Math.exp(im));
	} else {
		tmp = 0.041666666666666664 * (Math.sin(re) * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 120.0:
		tmp = (0.5 * math.sin(re)) * (2.0 + (im * im))
	elif im <= 1.16e+77:
		tmp = 0.5 * (math.exp(-im) + math.exp(im))
	else:
		tmp = 0.041666666666666664 * (math.sin(re) * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 120.0)
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.16e+77)
		tmp = Float64(0.5 * Float64(exp(Float64(-im)) + exp(im)));
	else
		tmp = Float64(0.041666666666666664 * Float64(sin(re) * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 120.0)
		tmp = (0.5 * sin(re)) * (2.0 + (im * im));
	elseif (im <= 1.16e+77)
		tmp = 0.5 * (exp(-im) + exp(im));
	else
		tmp = 0.041666666666666664 * (sin(re) * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 120.0], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.16e+77], N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.041666666666666664 * N[(N[Sin[re], $MachinePrecision] * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 120

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 84.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

   6. Simplified 84.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   if 120 < im < 1.1600000000000001e77

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin re\right)\right)}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   7. Applied egg-rr 56.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{\frac{\sin re}{\sin re + \left(\sin re - \sin re\right)}}\right) \cdot \left(e^{-im} + e^{im}\right) \]
   8. Step-by-step derivation

      1. +-inverses 56.3%

         \[\leadsto \left(0.5 \cdot \frac{\sin re}{\sin re + \color{blue}{0}}\right) \cdot \left(e^{-im} + e^{im}\right) \]

      2. +-rgt-identity 56.3%

         \[\leadsto \left(0.5 \cdot \frac{\sin re}{\color{blue}{\sin re}}\right) \cdot \left(e^{-im} + e^{im}\right) \]

      3. *-inverses 56.3%

         \[\leadsto \left(0.5 \cdot \color{blue}{1}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   9. Simplified 56.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{1}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   if 1.1600000000000001e77 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   6. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto 0.041666666666666664 \cdot \color{blue}{\left({im}^{4} \cdot \sin re\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left({im}^{4} \cdot \sin re\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 120:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(\sin re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 6: 81.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ t_1 := 2 + im \cdot im\\ \mathbf{if}\;im \leq 170000000:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+58}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;\left(t_1 + {im}^{4} \cdot 0.08333333333333333\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))) (t_1 (+ 2.0 (* im im))))
   (if (<= im 170000000.0)
     (* t_0 t_1)
     (if (<= im 4.3e+58)
       (pow re -512.0)
       (if (<= im 2.1e+153)
         (* (+ t_1 (* (pow im 4.0) 0.08333333333333333)) (* 0.5 re))
         (* t_0 (* im im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double t_1 = 2.0 + (im * im);
	double tmp;
	if (im <= 170000000.0) {
		tmp = t_0 * t_1;
	} else if (im <= 4.3e+58) {
		tmp = pow(re, -512.0);
	} else if (im <= 2.1e+153) {
		tmp = (t_1 + (pow(im, 4.0) * 0.08333333333333333)) * (0.5 * re);
	} else {
		tmp = t_0 * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    t_1 = 2.0d0 + (im * im)
    if (im <= 170000000.0d0) then
        tmp = t_0 * t_1
    else if (im <= 4.3d+58) then
        tmp = re ** (-512.0d0)
    else if (im <= 2.1d+153) then
        tmp = (t_1 + ((im ** 4.0d0) * 0.08333333333333333d0)) * (0.5d0 * re)
    else
        tmp = t_0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double t_1 = 2.0 + (im * im);
	double tmp;
	if (im <= 170000000.0) {
		tmp = t_0 * t_1;
	} else if (im <= 4.3e+58) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= 2.1e+153) {
		tmp = (t_1 + (Math.pow(im, 4.0) * 0.08333333333333333)) * (0.5 * re);
	} else {
		tmp = t_0 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	t_1 = 2.0 + (im * im)
	tmp = 0
	if im <= 170000000.0:
		tmp = t_0 * t_1
	elif im <= 4.3e+58:
		tmp = math.pow(re, -512.0)
	elif im <= 2.1e+153:
		tmp = (t_1 + (math.pow(im, 4.0) * 0.08333333333333333)) * (0.5 * re)
	else:
		tmp = t_0 * (im * im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	t_1 = Float64(2.0 + Float64(im * im))
	tmp = 0.0
	if (im <= 170000000.0)
		tmp = Float64(t_0 * t_1);
	elseif (im <= 4.3e+58)
		tmp = re ^ -512.0;
	elseif (im <= 2.1e+153)
		tmp = Float64(Float64(t_1 + Float64((im ^ 4.0) * 0.08333333333333333)) * Float64(0.5 * re));
	else
		tmp = Float64(t_0 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	t_1 = 2.0 + (im * im);
	tmp = 0.0;
	if (im <= 170000000.0)
		tmp = t_0 * t_1;
	elseif (im <= 4.3e+58)
		tmp = re ^ -512.0;
	elseif (im <= 2.1e+153)
		tmp = (t_1 + ((im ^ 4.0) * 0.08333333333333333)) * (0.5 * re);
	else
		tmp = t_0 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 170000000.0], N[(t$95$0 * t$95$1), $MachinePrecision], If[LessEqual[im, 4.3e+58], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, 2.1e+153], N[(N[(t$95$1 + N[(N[Power[im, 4.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.7e8

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 84.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

   6. Simplified 84.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   if 1.7e8 < im < 4.29999999999999991e58

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in re around 0 84.6%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   6. Applied egg-rr 31.3%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if 4.29999999999999991e58 < im < 2.10000000000000017e153

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in re around 0 63.6%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   5. Taylor expanded in im around 0 55.1%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   7. Simplified 55.1%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   if 2.10000000000000017e153 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

   6. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot {im}^{2}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 \cdot \sin re\right)} \]

      3. unpow2 100.0%

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 \cdot \sin re\right) \]

      4. *-commutative 100.0%

         \[\leadsto \left(im \cdot im\right) \cdot \color{blue}{\left(\sin re \cdot 0.5\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 170000000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 4.3 \cdot 10^{+58}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right) \cdot \left(0.5 \cdot re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 7: 66.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 540000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.95 \cdot 10^{+57}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+222}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 540000000.0)
   (sin re)
   (if (<= im 2.95e+57)
     (pow re -512.0)
     (if (<= im 4e+222)
       (* 0.041666666666666664 (* re (pow im 4.0)))
       (* 0.5 (* im (* (sin re) im)))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 540000000.0) {
		tmp = sin(re);
	} else if (im <= 2.95e+57) {
		tmp = pow(re, -512.0);
	} else if (im <= 4e+222) {
		tmp = 0.041666666666666664 * (re * pow(im, 4.0));
	} else {
		tmp = 0.5 * (im * (sin(re) * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 540000000.0d0) then
        tmp = sin(re)
    else if (im <= 2.95d+57) then
        tmp = re ** (-512.0d0)
    else if (im <= 4d+222) then
        tmp = 0.041666666666666664d0 * (re * (im ** 4.0d0))
    else
        tmp = 0.5d0 * (im * (sin(re) * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 540000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 2.95e+57) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= 4e+222) {
		tmp = 0.041666666666666664 * (re * Math.pow(im, 4.0));
	} else {
		tmp = 0.5 * (im * (Math.sin(re) * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 540000000.0:
		tmp = math.sin(re)
	elif im <= 2.95e+57:
		tmp = math.pow(re, -512.0)
	elif im <= 4e+222:
		tmp = 0.041666666666666664 * (re * math.pow(im, 4.0))
	else:
		tmp = 0.5 * (im * (math.sin(re) * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 540000000.0)
		tmp = sin(re);
	elseif (im <= 2.95e+57)
		tmp = re ^ -512.0;
	elseif (im <= 4e+222)
		tmp = Float64(0.041666666666666664 * Float64(re * (im ^ 4.0)));
	else
		tmp = Float64(0.5 * Float64(im * Float64(sin(re) * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 540000000.0)
		tmp = sin(re);
	elseif (im <= 2.95e+57)
		tmp = re ^ -512.0;
	elseif (im <= 4e+222)
		tmp = 0.041666666666666664 * (re * (im ^ 4.0));
	else
		tmp = 0.5 * (im * (sin(re) * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 540000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 2.95e+57], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, 4e+222], N[(0.041666666666666664 * N[(re * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(N[Sin[re], $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 5.4e8

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 69.6%

      \[\leadsto \color{blue}{\sin re} \]

   if 5.4e8 < im < 2.95000000000000006e57

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in re around 0 84.6%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   6. Applied egg-rr 31.3%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if 2.95000000000000006e57 < im < 4.0000000000000002e222

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in re around 0 70.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   5. Taylor expanded in im around 0 66.9%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   7. Simplified 66.9%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   8. Taylor expanded in im around inf 66.9%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)} \]

   if 4.0000000000000002e222 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

   6. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin re\right)\right)}\right) \cdot \left(2 + im \cdot im\right) \]

   9. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot {im}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left({im}^{2} \cdot \sin re\right)} \]

       2. unpow2 100.0%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(im \cdot im\right)} \cdot \sin re\right) \]

       3. associate-*l* 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \left(im \cdot \sin re\right)\right)} \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(im \cdot \left(im \cdot \sin re\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 540000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 2.95 \cdot 10^{+57}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 4 \cdot 10^{+222}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\sin re \cdot im\right)\right)\\ \end{array} \]

Alternative 8: 69.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 170000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5.7 \cdot 10^{+56}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 170000000.0)
   (sin re)
   (if (<= im 5.7e+56)
     (pow re -512.0)
     (if (<= im 2.1e+153)
       (* 0.041666666666666664 (* re (pow im 4.0)))
       (* (* 0.5 (sin re)) (* im im))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 170000000.0) {
		tmp = sin(re);
	} else if (im <= 5.7e+56) {
		tmp = pow(re, -512.0);
	} else if (im <= 2.1e+153) {
		tmp = 0.041666666666666664 * (re * pow(im, 4.0));
	} else {
		tmp = (0.5 * sin(re)) * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 170000000.0d0) then
        tmp = sin(re)
    else if (im <= 5.7d+56) then
        tmp = re ** (-512.0d0)
    else if (im <= 2.1d+153) then
        tmp = 0.041666666666666664d0 * (re * (im ** 4.0d0))
    else
        tmp = (0.5d0 * sin(re)) * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 170000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 5.7e+56) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= 2.1e+153) {
		tmp = 0.041666666666666664 * (re * Math.pow(im, 4.0));
	} else {
		tmp = (0.5 * Math.sin(re)) * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 170000000.0:
		tmp = math.sin(re)
	elif im <= 5.7e+56:
		tmp = math.pow(re, -512.0)
	elif im <= 2.1e+153:
		tmp = 0.041666666666666664 * (re * math.pow(im, 4.0))
	else:
		tmp = (0.5 * math.sin(re)) * (im * im)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 170000000.0)
		tmp = sin(re);
	elseif (im <= 5.7e+56)
		tmp = re ^ -512.0;
	elseif (im <= 2.1e+153)
		tmp = Float64(0.041666666666666664 * Float64(re * (im ^ 4.0)));
	else
		tmp = Float64(Float64(0.5 * sin(re)) * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 170000000.0)
		tmp = sin(re);
	elseif (im <= 5.7e+56)
		tmp = re ^ -512.0;
	elseif (im <= 2.1e+153)
		tmp = 0.041666666666666664 * (re * (im ^ 4.0));
	else
		tmp = (0.5 * sin(re)) * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 170000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 5.7e+56], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, 2.1e+153], N[(0.041666666666666664 * N[(re * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.7e8

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 69.6%

      \[\leadsto \color{blue}{\sin re} \]

   if 1.7e8 < im < 5.7000000000000002e56

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in re around 0 84.6%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   6. Applied egg-rr 31.3%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if 5.7000000000000002e56 < im < 2.10000000000000017e153

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in re around 0 63.6%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   5. Taylor expanded in im around 0 55.1%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   7. Simplified 55.1%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   8. Taylor expanded in im around inf 55.1%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)} \]

   if 2.10000000000000017e153 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

   6. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot {im}^{2}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 \cdot \sin re\right)} \]

      3. unpow2 100.0%

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 \cdot \sin re\right) \]

      4. *-commutative 100.0%

         \[\leadsto \left(im \cdot im\right) \cdot \color{blue}{\left(\sin re \cdot 0.5\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 170000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 5.7 \cdot 10^{+56}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 9: 81.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sin re\\ \mathbf{if}\;im \leq 170000000:\\ \;\;\;\;t_0 \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+55}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(im \cdot im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sin re))))
   (if (<= im 170000000.0)
     (* t_0 (+ 2.0 (* im im)))
     (if (<= im 1.15e+55)
       (pow re -512.0)
       (if (<= im 2.1e+153)
         (* 0.041666666666666664 (* re (pow im 4.0)))
         (* t_0 (* im im)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sin(re);
	double tmp;
	if (im <= 170000000.0) {
		tmp = t_0 * (2.0 + (im * im));
	} else if (im <= 1.15e+55) {
		tmp = pow(re, -512.0);
	} else if (im <= 2.1e+153) {
		tmp = 0.041666666666666664 * (re * pow(im, 4.0));
	} else {
		tmp = t_0 * (im * im);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sin(re)
    if (im <= 170000000.0d0) then
        tmp = t_0 * (2.0d0 + (im * im))
    else if (im <= 1.15d+55) then
        tmp = re ** (-512.0d0)
    else if (im <= 2.1d+153) then
        tmp = 0.041666666666666664d0 * (re * (im ** 4.0d0))
    else
        tmp = t_0 * (im * im)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sin(re);
	double tmp;
	if (im <= 170000000.0) {
		tmp = t_0 * (2.0 + (im * im));
	} else if (im <= 1.15e+55) {
		tmp = Math.pow(re, -512.0);
	} else if (im <= 2.1e+153) {
		tmp = 0.041666666666666664 * (re * Math.pow(im, 4.0));
	} else {
		tmp = t_0 * (im * im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sin(re)
	tmp = 0
	if im <= 170000000.0:
		tmp = t_0 * (2.0 + (im * im))
	elif im <= 1.15e+55:
		tmp = math.pow(re, -512.0)
	elif im <= 2.1e+153:
		tmp = 0.041666666666666664 * (re * math.pow(im, 4.0))
	else:
		tmp = t_0 * (im * im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sin(re))
	tmp = 0.0
	if (im <= 170000000.0)
		tmp = Float64(t_0 * Float64(2.0 + Float64(im * im)));
	elseif (im <= 1.15e+55)
		tmp = re ^ -512.0;
	elseif (im <= 2.1e+153)
		tmp = Float64(0.041666666666666664 * Float64(re * (im ^ 4.0)));
	else
		tmp = Float64(t_0 * Float64(im * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sin(re);
	tmp = 0.0;
	if (im <= 170000000.0)
		tmp = t_0 * (2.0 + (im * im));
	elseif (im <= 1.15e+55)
		tmp = re ^ -512.0;
	elseif (im <= 2.1e+153)
		tmp = 0.041666666666666664 * (re * (im ^ 4.0));
	else
		tmp = t_0 * (im * im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, 170000000.0], N[(t$95$0 * N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.15e+55], N[Power[re, -512.0], $MachinePrecision], If[LessEqual[im, 2.1e+153], N[(0.041666666666666664 * N[(re * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(im * im), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < 1.7e8

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 84.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

   6. Simplified 84.2%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   if 1.7e8 < im < 1.14999999999999994e55

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in re around 0 84.6%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   6. Applied egg-rr 31.3%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if 1.14999999999999994e55 < im < 2.10000000000000017e153

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in re around 0 63.6%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   5. Taylor expanded in im around 0 55.1%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   7. Simplified 55.1%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   8. Taylor expanded in im around inf 55.1%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)} \]

   if 2.10000000000000017e153 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

   6. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   7. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot {im}^{2}} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 \cdot \sin re\right)} \]

      3. unpow2 100.0%

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 \cdot \sin re\right) \]

      4. *-commutative 100.0%

         \[\leadsto \left(im \cdot im\right) \cdot \color{blue}{\left(\sin re \cdot 0.5\right)} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 170000000:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(2 + im \cdot im\right)\\ \mathbf{elif}\;im \leq 1.15 \cdot 10^{+55}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{+153}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(im \cdot im\right)\\ \end{array} \]

Alternative 10: 66.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 170000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 9.8 \cdot 10^{+58}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 170000000.0)
   (sin re)
   (if (<= im 9.8e+58)
     (pow re -512.0)
     (* 0.041666666666666664 (* re (pow im 4.0))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 170000000.0) {
		tmp = sin(re);
	} else if (im <= 9.8e+58) {
		tmp = pow(re, -512.0);
	} else {
		tmp = 0.041666666666666664 * (re * pow(im, 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 170000000.0d0) then
        tmp = sin(re)
    else if (im <= 9.8d+58) then
        tmp = re ** (-512.0d0)
    else
        tmp = 0.041666666666666664d0 * (re * (im ** 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 170000000.0) {
		tmp = Math.sin(re);
	} else if (im <= 9.8e+58) {
		tmp = Math.pow(re, -512.0);
	} else {
		tmp = 0.041666666666666664 * (re * Math.pow(im, 4.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 170000000.0:
		tmp = math.sin(re)
	elif im <= 9.8e+58:
		tmp = math.pow(re, -512.0)
	else:
		tmp = 0.041666666666666664 * (re * math.pow(im, 4.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 170000000.0)
		tmp = sin(re);
	elseif (im <= 9.8e+58)
		tmp = re ^ -512.0;
	else
		tmp = Float64(0.041666666666666664 * Float64(re * (im ^ 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 170000000.0)
		tmp = sin(re);
	elseif (im <= 9.8e+58)
		tmp = re ^ -512.0;
	else
		tmp = 0.041666666666666664 * (re * (im ^ 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 170000000.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 9.8e+58], N[Power[re, -512.0], $MachinePrecision], N[(0.041666666666666664 * N[(re * N[Power[im, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 1.7e8

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 69.6%

      \[\leadsto \color{blue}{\sin re} \]

   if 1.7e8 < im < 9.80000000000000037e58

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in re around 0 84.6%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   6. Applied egg-rr 31.3%

      \[\leadsto \color{blue}{{re}^{-512}} \]

   if 9.80000000000000037e58 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in re around 0 68.0%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(e^{-im} + e^{im}\right) \]

   5. Taylor expanded in im around 0 66.1%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(2 + \left({im}^{2} + 0.08333333333333333 \cdot {im}^{4}\right)\right)} \]

   7. Simplified 66.1%

      \[\leadsto \left(0.5 \cdot re\right) \cdot \color{blue}{\left(\left(2 + im \cdot im\right) + {im}^{4} \cdot 0.08333333333333333\right)} \]

   8. Taylor expanded in im around inf 66.1%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 170000000:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 9.8 \cdot 10^{+58}:\\ \;\;\;\;{re}^{-512}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \left(re \cdot {im}^{4}\right)\\ \end{array} \]

Alternative 11: 63.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 760:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+148}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 760.0)
   (sin re)
   (if (<= im 6.5e+148)
     (+
      0.08333333333333333
      (+ (/ 0.25 (* re re)) (* (* re re) 0.016666666666666666)))
     (* (+ 2.0 (* im im)) (* 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 760.0) {
		tmp = sin(re);
	} else if (im <= 6.5e+148) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else {
		tmp = (2.0 + (im * im)) * (0.5 * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 760.0d0) then
        tmp = sin(re)
    else if (im <= 6.5d+148) then
        tmp = 0.08333333333333333d0 + ((0.25d0 / (re * re)) + ((re * re) * 0.016666666666666666d0))
    else
        tmp = (2.0d0 + (im * im)) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 760.0) {
		tmp = Math.sin(re);
	} else if (im <= 6.5e+148) {
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	} else {
		tmp = (2.0 + (im * im)) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 760.0:
		tmp = math.sin(re)
	elif im <= 6.5e+148:
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666))
	else:
		tmp = (2.0 + (im * im)) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 760.0)
		tmp = sin(re);
	elseif (im <= 6.5e+148)
		tmp = Float64(0.08333333333333333 + Float64(Float64(0.25 / Float64(re * re)) + Float64(Float64(re * re) * 0.016666666666666666)));
	else
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 760.0)
		tmp = sin(re);
	elseif (im <= 6.5e+148)
		tmp = 0.08333333333333333 + ((0.25 / (re * re)) + ((re * re) * 0.016666666666666666));
	else
		tmp = (2.0 + (im * im)) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 760.0], N[Sin[re], $MachinePrecision], If[LessEqual[im, 6.5e+148], N[(0.08333333333333333 + N[(N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 760

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 70.0%

      \[\leadsto \color{blue}{\sin re} \]

   if 760 < im < 6.49999999999999947e148

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   5. Applied egg-rr 10.2%

      \[\leadsto \color{blue}{{\left(\sin re \cdot -2\right)}^{-2}} \]

   6. Taylor expanded in re around 0 29.7%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.25 \cdot \frac{1}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 29.7%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      2. metadata-eval 29.7%

         \[\leadsto 0.08333333333333333 + \left(\frac{\color{blue}{0.25}}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      3. unpow2 29.7%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{\color{blue}{re \cdot re}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      4. *-commutative 29.7%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{{re}^{2} \cdot 0.016666666666666666}\right) \]

      5. unpow2 29.7%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666\right) \]

   8. Simplified 29.7%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]

   if 6.49999999999999947e148 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

   6. Simplified 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   7. Taylor expanded in re around 0 69.2%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(2 + im \cdot im\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 760:\\ \;\;\;\;\sin re\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{+148}:\\ \;\;\;\;0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 12: 37.6% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 400:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 2.25 \cdot 10^{+105}:\\ \;\;\;\;0.08333333333333333 + re \cdot \left(re \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 400.0)
   re
   (if (<= im 2.25e+105)
     (+ 0.08333333333333333 (* re (* re 0.016666666666666666)))
     (* (* im im) (* 0.5 re)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 400.0) {
		tmp = re;
	} else if (im <= 2.25e+105) {
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	} else {
		tmp = (im * im) * (0.5 * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 400.0d0) then
        tmp = re
    else if (im <= 2.25d+105) then
        tmp = 0.08333333333333333d0 + (re * (re * 0.016666666666666666d0))
    else
        tmp = (im * im) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 400.0) {
		tmp = re;
	} else if (im <= 2.25e+105) {
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	} else {
		tmp = (im * im) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 400.0:
		tmp = re
	elif im <= 2.25e+105:
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666))
	else:
		tmp = (im * im) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 400.0)
		tmp = re;
	elseif (im <= 2.25e+105)
		tmp = Float64(0.08333333333333333 + Float64(re * Float64(re * 0.016666666666666666)));
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 400.0)
		tmp = re;
	elseif (im <= 2.25e+105)
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	else
		tmp = (im * im) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 400.0], re, If[LessEqual[im, 2.25e+105], N[(0.08333333333333333 + N[(re * N[(re * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(im * im), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if im < 400

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 70.0%

      \[\leadsto \color{blue}{\sin re} \]

   5. Taylor expanded in re around 0 38.7%

      \[\leadsto \color{blue}{re} \]

   if 400 < im < 2.2500000000000001e105

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   5. Applied egg-rr 12.5%

      \[\leadsto \color{blue}{{\left(\sin re \cdot -2\right)}^{-2}} \]

   6. Taylor expanded in re around 0 36.8%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.25 \cdot \frac{1}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 36.8%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      2. metadata-eval 36.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{\color{blue}{0.25}}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      3. unpow2 36.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{\color{blue}{re \cdot re}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      4. *-commutative 36.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{{re}^{2} \cdot 0.016666666666666666}\right) \]

      5. unpow2 36.8%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666\right) \]

   8. Simplified 36.8%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]

   9. Taylor expanded in re around inf 26.5%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.016666666666666666 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 26.5%

          \[\leadsto 0.08333333333333333 + 0.016666666666666666 \cdot \color{blue}{\left(re \cdot re\right)} \]

       2. *-commutative 26.5%

          \[\leadsto 0.08333333333333333 + \color{blue}{\left(re \cdot re\right) \cdot 0.016666666666666666} \]

       3. associate-*r* 26.5%

          \[\leadsto 0.08333333333333333 + \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)} \]

   11. Simplified 26.5%

       \[\leadsto \color{blue}{0.08333333333333333 + re \cdot \left(re \cdot 0.016666666666666666\right)} \]

   if 2.2500000000000001e105 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 89.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

   6. Simplified 89.3%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   7. Taylor expanded in im around inf 89.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 89.3%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot {im}^{2}} \]

      2. *-commutative 89.3%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 \cdot \sin re\right)} \]

      3. unpow2 89.3%

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 \cdot \sin re\right) \]

      4. *-commutative 89.3%

         \[\leadsto \left(im \cdot im\right) \cdot \color{blue}{\left(\sin re \cdot 0.5\right)} \]

   9. Simplified 89.3%

      \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)} \]

   10. Taylor expanded in re around 0 63.8%

       \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{re} \cdot 0.5\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 400:\\ \;\;\;\;re\\ \mathbf{elif}\;im \leq 2.25 \cdot 10^{+105}:\\ \;\;\;\;0.08333333333333333 + re \cdot \left(re \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 13: 48.2% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.45 \cdot 10^{+159}:\\ \;\;\;\;0.08333333333333333 + re \cdot \left(re \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.45e+159)
   (+ 0.08333333333333333 (* re (* re 0.016666666666666666)))
   (* (+ 2.0 (* im im)) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.45e+159) {
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	} else {
		tmp = (2.0 + (im * im)) * (0.5 * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.45d+159)) then
        tmp = 0.08333333333333333d0 + (re * (re * 0.016666666666666666d0))
    else
        tmp = (2.0d0 + (im * im)) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.45e+159) {
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	} else {
		tmp = (2.0 + (im * im)) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.45e+159:
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666))
	else:
		tmp = (2.0 + (im * im)) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.45e+159)
		tmp = Float64(0.08333333333333333 + Float64(re * Float64(re * 0.016666666666666666)));
	else
		tmp = Float64(Float64(2.0 + Float64(im * im)) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.45e+159)
		tmp = 0.08333333333333333 + (re * (re * 0.016666666666666666));
	else
		tmp = (2.0 + (im * im)) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.45e+159], N[(0.08333333333333333 + N[(re * N[(re * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(im * im), $MachinePrecision]), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.4500000000000001e159

   1. Initial program 99.9%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 99.9%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   5. Applied egg-rr 7.9%

      \[\leadsto \color{blue}{{\left(\sin re \cdot -2\right)}^{-2}} \]

   6. Taylor expanded in re around 0 33.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(0.25 \cdot \frac{1}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 33.5%

         \[\leadsto 0.08333333333333333 + \left(\color{blue}{\frac{0.25 \cdot 1}{{re}^{2}}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      2. metadata-eval 33.5%

         \[\leadsto 0.08333333333333333 + \left(\frac{\color{blue}{0.25}}{{re}^{2}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      3. unpow2 33.5%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{\color{blue}{re \cdot re}} + 0.016666666666666666 \cdot {re}^{2}\right) \]

      4. *-commutative 33.5%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{{re}^{2} \cdot 0.016666666666666666}\right) \]

      5. unpow2 33.5%

         \[\leadsto 0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \color{blue}{\left(re \cdot re\right)} \cdot 0.016666666666666666\right) \]

   8. Simplified 33.5%

      \[\leadsto \color{blue}{0.08333333333333333 + \left(\frac{0.25}{re \cdot re} + \left(re \cdot re\right) \cdot 0.016666666666666666\right)} \]

   9. Taylor expanded in re around inf 33.5%

      \[\leadsto \color{blue}{0.08333333333333333 + 0.016666666666666666 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 33.5%

          \[\leadsto 0.08333333333333333 + 0.016666666666666666 \cdot \color{blue}{\left(re \cdot re\right)} \]

       2. *-commutative 33.5%

          \[\leadsto 0.08333333333333333 + \color{blue}{\left(re \cdot re\right) \cdot 0.016666666666666666} \]

       3. associate-*r* 33.5%

          \[\leadsto 0.08333333333333333 + \color{blue}{re \cdot \left(re \cdot 0.016666666666666666\right)} \]

   11. Simplified 33.5%

       \[\leadsto \color{blue}{0.08333333333333333 + re \cdot \left(re \cdot 0.016666666666666666\right)} \]

   if -3.4500000000000001e159 < re

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 79.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

   6. Simplified 79.6%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   7. Taylor expanded in re around 0 56.3%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(2 + im \cdot im\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.45 \cdot 10^{+159}:\\ \;\;\;\;0.08333333333333333 + re \cdot \left(re \cdot 0.016666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 14: 37.3% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 170000000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 170000000.0) re (* (* im im) (* 0.5 re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 170000000.0) {
		tmp = re;
	} else {
		tmp = (im * im) * (0.5 * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 170000000.0d0) then
        tmp = re
    else
        tmp = (im * im) * (0.5d0 * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 170000000.0) {
		tmp = re;
	} else {
		tmp = (im * im) * (0.5 * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 170000000.0:
		tmp = re
	else:
		tmp = (im * im) * (0.5 * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 170000000.0)
		tmp = re;
	else
		tmp = Float64(Float64(im * im) * Float64(0.5 * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 170000000.0)
		tmp = re;
	else
		tmp = (im * im) * (0.5 * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 170000000.0], re, N[(N[(im * im), $MachinePrecision] * N[(0.5 * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.7e8

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 69.6%

      \[\leadsto \color{blue}{\sin re} \]

   5. Taylor expanded in re around 0 38.5%

      \[\leadsto \color{blue}{re} \]

   if 1.7e8 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 63.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + {im}^{2}\right)} \]

   6. Simplified 63.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(2 + im \cdot im\right)} \]

   7. Taylor expanded in im around inf 63.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin re \cdot {im}^{2}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 63.4%

         \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot {im}^{2}} \]

      2. *-commutative 63.4%

         \[\leadsto \color{blue}{{im}^{2} \cdot \left(0.5 \cdot \sin re\right)} \]

      3. unpow2 63.4%

         \[\leadsto \color{blue}{\left(im \cdot im\right)} \cdot \left(0.5 \cdot \sin re\right) \]

      4. *-commutative 63.4%

         \[\leadsto \left(im \cdot im\right) \cdot \color{blue}{\left(\sin re \cdot 0.5\right)} \]

   9. Simplified 63.4%

      \[\leadsto \color{blue}{\left(im \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)} \]

   10. Taylor expanded in re around 0 45.4%

       \[\leadsto \left(im \cdot im\right) \cdot \left(\color{blue}{re} \cdot 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 170000000:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot im\right) \cdot \left(0.5 \cdot re\right)\\ \end{array} \]

Alternative 15: 29.5% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 980:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= im 980.0) re (/ 0.25 (* re re))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 980.0) {
		tmp = re;
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 980.0d0) then
        tmp = re
    else
        tmp = 0.25d0 / (re * re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 980.0) {
		tmp = re;
	} else {
		tmp = 0.25 / (re * re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 980.0:
		tmp = re
	else:
		tmp = 0.25 / (re * re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 980.0)
		tmp = re;
	else
		tmp = Float64(0.25 / Float64(re * re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 980.0)
		tmp = re;
	else
		tmp = 0.25 / (re * re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 980.0], re, N[(0.25 / N[(re * re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 980

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   4. Taylor expanded in im around 0 70.0%

      \[\leadsto \color{blue}{\sin re} \]

   5. Taylor expanded in re around 0 38.7%

      \[\leadsto \color{blue}{re} \]

   if 980 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   5. Applied egg-rr 12.7%

      \[\leadsto \color{blue}{{\left(\sin re \cdot -2\right)}^{-2}} \]

   6. Taylor expanded in re around 0 12.5%

      \[\leadsto \color{blue}{\frac{0.25}{{re}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 12.5%

         \[\leadsto \frac{0.25}{\color{blue}{re \cdot re}} \]

   8. Simplified 12.5%

      \[\leadsto \color{blue}{\frac{0.25}{re \cdot re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 980:\\ \;\;\;\;re\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{re \cdot re}\\ \end{array} \]

Alternative 16: 26.7% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ re \end{array} \]

FPCore

(FPCore (re im) :precision binary64 re)

C

double code(double re, double im) {
	return re;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = re
end function

Java

public static double code(double re, double im) {
	return re;
}

Python

def code(re, im):
	return re

Julia

function code(re, im)
	return re
end

MATLAB

function tmp = code(re, im)
	tmp = re;
end

Wolfram

code[re_, im_] := re

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 100.0%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(e^{\color{blue}{-im}} + e^{im}\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

4. Taylor expanded in im around 0 53.2%

   \[\leadsto \color{blue}{\sin re} \]

5. Taylor expanded in re around 0 29.6%

   \[\leadsto \color{blue}{re} \]

6. Final simplification 29.6%

   \[\leadsto re \]

Reproduce

herbie shell --seed 2023188 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))